Time dependent flows in arrested states

A three dimensional (3d) calculation of the mode coupling equations with integration through transients (MCT-ITT) under shear is now possible and fruitful. The outcome is, beyond linear response, completely anisotropic. The numerics remains demanding, but the ongoing progress on the hardware side will soon ease these problems.

The 3d results are in full accordance with earlier two-dimensional (2d) calculations, especially the appearance of a yield scaling (alpha master) function under shear. MCT-ITT also describes that particles circumvent each others easier in 3d, because correlator oscillations in the final decay region are less pronounced.

The MCT-ITT transient, coherent density correlator was compared with steady-state incoherent correlators of a MD simulation and a PMMA experiment. The qualitative agreement was high, however, a rescaling strain parameter of 10 to 30 had to be introduced, because MCT-ITT correlators decay too slow. Also the stress overshoot occurs retarded by a strain factor of about 3 to 5.

Shear and normal stresses were calculated to show the full tensorial and anisotropic nature of the theory. A connection between the peak strain gamma* of the stress overshoot and height with the flow curves was deduced from MCT's generalized shear modulus. The peak-strain vs shear-rate curves look qualitatively the same as the flow curves. This means that the peak strain increases with packing fraction and bare Péclet number. The overshoot vanishes in a fluid.

Within the full anisotropic calculations it turned out that the second normal stress is negative. The Lodge-Meissner relationship was proven to hold in MCT-ITT and a relationship to the first normal stress was derived, which is similar to polymeric melts.

Structure factor (SF) distortions calculated with MCT-ITT could be illustrated for the first time in 3d. A deformation of the SF according to the deformation field of simple shear flow was verified. This naturally causes a change from a quadrupolar symmetry in the elastic regime to a more hexadecapolar distortion in the steady state. The appearance of the stress overshoot is closely connected to this, because only quadrupolar distortions increase the shear stress, a change in symmetries decreases it. Motivated by these findings, a heuristic derivation of the Lindemann criterion from the MCT equations for the shear stress could be done with just using symmetry considerations and the quiescent SF.

A comparison of the shear-flow, vorticity plane of the SF distortion with experiments of X-ray scattering on silica particles yielded a good qualitative agreement. The experimental SF exhibits a contraction along the shear flow axis and so does the SF of MCT. A difference of a factor of 3 in the distortion amplitudes coincides well with the difference of MCT's peak strain compared to other colloidal experiments.

An anisotropic contribution to the intermediate small strain regime of mode coupling theory (MCT) under simple shear was derived theoretically and approved in comparison to the full numerical solution of MCT in two and three dimensions (2d and 3d). The anisotropic term is of quadrupolar symmetry. It was verified for the full numerics that the initial quadrupolar symmetry in the shear induced decay decreases with time. A tendency of the 3d correlators to decay at larger strains than in 2d was observed. Besides this, taking the third spacial dimension into account does not add new qualitative differences to the intermediate time regime, in which the beta-analysis is valid; the 3d correlators inherit the quadrupolar symmetry from the two dimensional shear-flow, shear-gradient plane.

A schematic model of mode coupling theory was generalized to describe consistently flow curves, linear response shear moduli and nonlinear stress-strain curves, especially the stress overshoot. The generalization holds in a universal, homogeneous straining geometry, with a focus on simple-shear and compressional flow. This was done by implementing a time-dependent vertex function in the generalized shear modulus, which automatically incorporates via the Finger tensor the strain geometry. Schematic MCT remains isotropic, but dependent on the rotational invariants of the Finger tensor, on which the isotropic vertex also solely depends.

The new vertex is in accordance with microscopic MCT as long as bare Péclet numbers are asymptotically small and the correlator decay is governed by strain. For higher Péclet numbers, model parameters must be varied to mimic a influence of alpha and beta decay on the new vertex. Effectively just one new rheological parameter, the peak strain gamma*, at which the stress is maximal, is added to the model, because a second parameter, turned out to be linearly coupled to gamma*. A comparison to experiments and MD simulations showed that microscopic MCT overestimates gamma* by a factor of 3-5.

Together with experiments on metallic glasses, the schematic model was applied to fit compressional flow. It turned out that the parameters of the model are universal to different flow geometries when a rescaled strain rate is defined. It explains why the peak strain is shifted to smaller values under compression, compared to shear flow. A comparison of the colloidal experiment with the metallic glasses via schematic MCT revealed that a universal structural behavior under strain in these very different systems can be identified and that it is captured by MCT's structural decay. Decay-strain and energy scales are of the same order when renormalized properly by flow geometry and thermal energy per particle size.

Single-particle mean square displacements could be described in using the (transient) viscosity of the schematic model in a generalized Stokes-Einstein relation. Describing the long-time limit of a MD simulation of hard spheres worked well this way. A superdiffusive motion in start-up shear is described by MCT just around the strain of the stress overshoot, which the simulation yielded as well.

The overall result is that qualitative agreement of MCT-ITT with experiments and simulations is so high that the structural physics must be described correctly by the theory.

2014-02-04T07:18:55ZA three dimensional (3d) calculation of the mode coupling equations with integration through transients (MCT-ITT) under shear is now possible and fruitful. The outcome is, beyond linear response, completely anisotropic. The numerics remains demanding, but the ongoing progress on the hardware side will soon ease these problems.<br /><br /><br /><br />The 3d results are in full accordance with earlier two-dimensional (2d) calculations, especially the appearance of a yield scaling (alpha master) function under shear. MCT-ITT also describes that particles circumvent each others easier in 3d, because correlator oscillations in the final decay region are less pronounced.<br /><br /><br /><br />The MCT-ITT transient, coherent density correlator was compared with steady-state incoherent correlators of a MD simulation and a PMMA experiment. The qualitative agreement was high, however, a rescaling strain parameter of 10 to 30 had to be introduced, because MCT-ITT correlators decay too slow. Also the stress overshoot occurs retarded by a strain factor of about 3 to 5.<br /><br /><br /><br />Shear and normal stresses were calculated to show the full tensorial and anisotropic nature of the theory. A connection between the peak strain gamma* of the stress overshoot and height with the flow curves was deduced from MCT's generalized shear modulus. The peak-strain vs shear-rate curves look qualitatively the same as the flow curves. This means that the peak strain increases with packing fraction and bare Péclet number. The overshoot vanishes in a fluid.<br /><br /><br /><br />Within the full anisotropic calculations it turned out that the second normal stress is negative. The Lodge-Meissner relationship was proven to hold in MCT-ITT and a relationship to the first normal stress was derived, which is similar to polymeric melts.<br /><br /><br /><br />Structure factor (SF) distortions calculated with MCT-ITT could be illustrated for the first time in 3d. A deformation of the SF according to the deformation field of simple shear flow was verified. This naturally causes a change from a quadrupolar symmetry in the elastic regime to a more hexadecapolar distortion in the steady state. The appearance of the stress overshoot is closely connected to this, because only quadrupolar distortions increase the shear stress, a change in symmetries decreases it. Motivated by these findings, a heuristic derivation of the Lindemann criterion from the MCT equations for the shear stress could be done with just using symmetry considerations and the quiescent SF.<br /><br /><br /><br />A comparison of the shear-flow, vorticity plane of the SF distortion with experiments of X-ray scattering on silica particles yielded a good qualitative agreement. The experimental SF exhibits a contraction along the shear flow axis and so does the SF of MCT. A difference of a factor of 3 in the distortion amplitudes coincides well with the difference of MCT's peak strain compared to other colloidal experiments.<br /><br /><br /><br />An anisotropic contribution to the intermediate small strain regime of mode coupling theory (MCT) under simple shear was derived theoretically and approved in comparison to the full numerical solution of MCT in two and three dimensions (2d and 3d). The anisotropic term is of quadrupolar symmetry. It was verified for the full numerics that the initial quadrupolar symmetry in the shear induced decay decreases with time. A tendency of the 3d correlators to decay at larger strains than in 2d was observed. Besides this, taking the third spacial dimension into account does not add new qualitative differences to the intermediate time regime, in which the beta-analysis is valid; the 3d correlators inherit the quadrupolar symmetry from the two dimensional shear-flow, shear-gradient plane.<br /><br /><br /><br />A schematic model of mode coupling theory was generalized to describe consistently flow curves, linear response shear moduli and nonlinear stress-strain curves, especially the stress overshoot. The generalization holds in a universal, homogeneous straining geometry, with a focus on simple-shear and compressional flow. This was done by implementing a time-dependent vertex function in the generalized shear modulus, which automatically incorporates via the Finger tensor the strain geometry. Schematic MCT remains isotropic, but dependent on the rotational invariants of the Finger tensor, on which the isotropic vertex also solely depends.<br /><br /><br /><br />The new vertex is in accordance with microscopic MCT as long as bare Péclet numbers are asymptotically small and the correlator decay is governed by strain. For higher Péclet numbers, model parameters must be varied to mimic a influence of alpha and beta decay on the new vertex. Effectively just one new rheological parameter, the peak strain gamma*, at which the stress is maximal, is added to the model, because a second parameter, turned out to be linearly coupled to gamma*. A comparison to experiments and MD simulations showed that microscopic MCT overestimates gamma* by a factor of 3-5.<br /><br /><br /><br />Together with experiments on metallic glasses, the schematic model was applied to fit compressional flow. It turned out that the parameters of the model are universal to different flow geometries when a rescaled strain rate is defined. It explains why the peak strain is shifted to smaller values under compression, compared to shear flow. A comparison of the colloidal experiment with the metallic glasses via schematic MCT revealed that a universal structural behavior under strain in these very different systems can be identified and that it is captured by MCT's structural decay. Decay-strain and energy scales are of the same order when renormalized properly by flow geometry and thermal energy per particle size.<br /><br /><br /><br />Single-particle mean square displacements could be described in using the (transient) viscosity of the schematic model in a generalized Stokes-Einstein relation. Describing the long-time limit of a MD simulation of hard spheres worked well this way. A superdiffusive motion in start-up shear is described by MCT just around the strain of the stress overshoot, which the simulation yielded as well.<br /><br /><br /><br />The overall result is that qualitative agreement of MCT-ITT with experiments and simulations is so high that the structural physics must be described correctly by the theory.engterms-of-useAmann, Christian Peter2013Amann, Christian Peter2014-02-04T07:18:55ZTime dependent flows in arrested states